Critical Points Of An Autoencoder Can Provably Recover Sparsely Used Overcomplete Dictionaries

In "Dictionary Learning" one is trying to recover incoherent matrices $A^* \in \mathbb{R}^{n \times h}$ (typically overcomplete and whose columns are assumed to be normalized) and sparse vectors $x^* \in \mathbb{R}^h$ with a small support of size $h^p$ for some $0 <p < 1$ while being given access to observations $y \in \mathbb{R}^n$ where $y = A^*x^*$. In this work we undertake a rigorous analysis of the possibility that dictionary learning could be performed by gradient descent on "Autoencoders", which are $\mathbb{R}^n \rightarrow \mathbb{R}^n$ neural network with a single ReLU activation layer of size $h$. Towards the above objective we propose a new autoencoder loss function which modifies the squared loss error term and also adds new regularization terms. We create a proxy for the expected gradient of this loss function which we motivate with high probability arguments, under natural distributional assumptions on the sparse code $x^*$. Under the same distributional assumptions on $x^*$, we show that, in the limit of large enough sparse code dimension, any zero point of our proxy for the expected gradient of the loss function within a certain radius of $A^*$ corresponds to dictionaries whose action on the sparse vectors is indistinguishable from that of $A^*$. We also report simulations on synthetic data in support of our theory.